Code Division

CDMA is a kind of multiple access technique that uses the different types of code used to transmit the signals to distinguish the different channels. It assigns a special address code for each user to distinguish the different channels, while each user can completely occupy the whole frequency and time resources.

From the upper plot we can see that for different users, different kinds of waveforms(codes) are used to transform the signal.

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Orthogonal

Assume that x = ( x₁, x₂, x_3, … x_n) and y = ( y₁, y₂, y₃, … y_n) are two codes with the same length and that the x₁, x₂, x_3, … x_n and y₁, y₂, y₃, … y_n with only values +1 and -1.

We define the correlation coefficient

If ρ(x,y)=0, we say that x and y are orthogonal.

For example, assume that S₁(t)=(+1,+1,+1,+1) and S₂(t)=(+1,+1,-1,-1).

(+1)* (+1)+ (+1)* (+1)+ (+1)*(-1)+ (+1)*(-1)=0

So they are orthogonal.

Theoretically, as long as the address codes we assign for each user are orthogonal and that the we use the same address code at the receiver as that used at the transmitter, we can completely distinguish the different channels even though the transmitted signals might overlap in frequency、time or space domain.

Specifically, we use the same assumption that S₁(t)=(+1,+1,+1,+1) and S₂(t)=(+1,+1,-1,-1). And since they are orthogonal, we can use them as the address code to distinguish the different channels.

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In the plot above, we assume that user A use the address code S₁(t)=(+1,+1,+1,+1) and user B use the address code S₂(t)=(+1,+1,-1,-1). At the transmitter, we multiply the signal we’d like to transmit by the address code to form the transmitted signal, and at the receiver we demodulate the signal by multiplying the received signal with the same address code and then summing.

Assume that for user A to transmit ‘a’, the modulated signal is a*(+1,+1,+1,+1) =(a,a,a,a); and for user B to transmit ‘b’, the modulated signal is b*(+1,+1,-1,-1) =(b,b,-b,-b). As the signal covers all the frequency、time and space in CDMA, both the two modulated signals appear at the receivers of A and B.

For A, the received signal (a,a,a,a)* (+1,+1,+1,+1)=(a,a,a,a), after summing we get ‘a’, and the received signal (b,b,-b,-b)* (+1,+1,+1,+1)= (b,b,-b,-b), after summing we get ‘0’.

For B, the received signal (a,a,a,a)* (+1,+1,-1,-1)=(a,a,-a,-a), after summing we get ‘0’, and the received signal (b,b,-b,-b)* (+1,+1,-1,-1)=(b,b,b,b), after summing we get ‘b’.

From the example above, we can see that as the address codes assigned for the two users are orthogonal, so the demodulated signal is ‘0’ if the signal is not modulated by the same address code as that at the receiver, which can be expressed by

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